If the insurance company issued you a check for the damages, either via estimates you submitted or by their own adjuster, and you choose this shop on your own then yes, the issue is between you and the shop.
Based on your post:
...shop still has my 4Runner and I've not paid them for it yet.
It sounds like this is the case so you may need to fight this with the shop directly.
However, if you took your car to one of those insurance company “approved” shops and the insurance company is paying the shop directly then you will need to fight this through the insurance company.
My first question is what’s “not good”? If the paint does not match or fenders / doors are misaligned, hoods crocked then you should have a solid basis to fight the shop. Take photos and document all your conversations as this will help if you litigate. If you are talking about the fact that they used old parts, e.g. used tail lights or head lights that are fading, then you may not get anywhere trying to fight this sort of thing.
It’s good that you have not paid the shop yet as this gives you some leverage in getting the work done right.
HOWEVER be careful about this! Depending on the laws in your state, if you don’t pay for repairs in a given period of time, the shop might have the right to take possession of your vehicle and sell it at auction to recover their losses. I am not saying this to try and scare you or anything, just be aware that most states allow for this sort of thing so make sure you understand what rights you have in your state and what laws apply here.
If you are unable to come to an agreement with the shop that did the repairs, I would suggest that you call your insurance agent and discuss the issue with them. Depending on the situation, you might be able to submit a “no fault” claim with your own insurance company for the damages. Usually when this happens your insurance will try and recover their loses by going after the other drivers insurance. But it all depends on the situation so it’s best that you talk with your own insurance agent for advice.
